Laplacian solution for a quasi-toroidal electrode configuration suitable for experiments in dielectrophoresis and thermoelectroconvection

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ELECTROSTATICS ELSEVIER

Journal of Electrostatics 35 (1995) 323 338

Laplacian solution for a quasi-toroidal electrode configuration suitable for experiments in dielectrophoresis and thermoelectroconvection H.W. Vanderschueren a'*, M.-O. Louppe b, A.

Vanderschueren

a

"University of Libge, lnstitut Monteflore, B28 Sart-Tihnan, B-4000 Likge, Belgium h DIC Coll., rue Sur-la-Fontaine 70, or Coll. SB-SS, rue St.-Gilles, B-4000 Libge. Belgium

Received 17 May 1994; accepted after revision 2 February 1995

Abstract Since the discovery of the thermoelectroconvection effect (i.e. the effect of an electric field on heat transfer in fluids) by Senftleben in 1931, numerous studies have been made with the electrode configuration he used himself, namely a thin metal wire mounted in the axis of a cylinder. We propose here the theoretical treatment of another configuration which proved to be convenient, namely a wire in the axis of a ring.This configuration gives nearly the same field in the vicinity of the wire as in the cylindrical problem, "transparency" of the system allowing optical observation (by Schlieren optics) and, last but not least, nearly complete analytical treatment valid without restriction in the central part of the cell.

1. Introduction W h e n Senftleben decided to study experimentally the effect of the electric field on the thermal convection in a gas [1], he chose - quite naturally - the cylindrical configuration as being the simplest one to work with. After him, it seemed natural for the researchers w h o followed his tracks and extended the studies to the liquids to a d o p t the same system which is also classic in simple dielectrophoresis [2]. However, at a glance, it is evident that this form of cell is inherently limited in at least two respects. First, it is only possible to observe the p h e n o m e n a radially by Schlieren optics or interferometry since the cylinder surrounds the heated wire. Second, the cylinder constitutes an obstacle to the simple development of the fluid motions. As the wire is e m b e d d e d in a fluid contained in a quasi closed cell - the cylinder - different m o t i o n s can take place which are directly conditioned by the

* Corresponding author. E-mail: [email protected]. Fax: 3241662877. Tel.: 3241662670. 0304-3886/95/$09.50 :~ 1995 Elsevier Science B.V. All rights reserved SSD1 0 3 0 4 - 3 8 8 6 ( 9 5 ) 0 0 0 0 5 - 4

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H.W. Vanderschueren et al./Journal of Electrostatics 35 (1995) 323-338

Fig. 1. A model of warm fluid movementwhich can evolvetowards a quasi-stable one or degenerateinto a turbulent one according to the initial conditions in a cylindricalceil.

latter. The warm fluid arising from the wire accumulates at the top of the cylinder which can influence the further conditions, for instance by modifying the permittivity, on the one hand, and generate permanent motions in the whole cylinder as evoked in Fig. 1 on the other. These latter motions are in themselves very interesting to study but, for investigating electroconvection or dielectrophoresis, it is highly desirable to decouple the effects in order to overcome such complex domains where artefacts are not rare and can perturb the results in question [3]. Some time ago, we proposed a promising new configuration which makes up for the above-mentioned deficiencies [4] and has been exploited in various circumstances up till now [5-8]. An appreciable additional interest of our configuration is the possibility to allow a complete analytical electrostatic treatment by means of an approximation which preserves the integrity of the system in the central part of the whole cell, i.e. including the wire and the opposite electrode as well as the embedding fluid. The basic idea was presented previously by two of us [9] but at a time when the computational means were too limited for easily exploiting the model. The main purpose of this paper is to give the fundamental idea of the approach and the subsequent simple mathematical treatment with justification of the approximation.

2. Experimental design Fig. 2 shows the configuration we propose to study here. The central electrode is the same as that used by Senftleben and most subsequent workers, namely a platinum wire heated by Joule effect. However, the opposite electrode is a ring coaxially disposed around the wire in the median plane. Some of the advantages previously mentioned are immediately perceptible. The cell is completely transparent and

H.W. Vanderschueren et al./Journal of Electrostatics 35 (1995) 323 338

current

3125

ring

sense Fig. 2. Schematicdesign of the electrode geometry.The diameter of the platinum wire is currently 25 ~m. The diameter of the ring is commonlyof the order of 30 mm and its thickness is of the order of I mm.

Fig. 3. A priori evident potential repartition along a radius in the central ring plane.

consequently can be observed by Schlieren optics or by interferometry for example. In the central part of the cell, i.e. near the plane of the ring, close to the wire, we have nearly the same electrostatic field distribution as in the classical cylindrical system for a wire thin compared to the ring diameter. At a certain distance, on the contrary, a second field divergence is induced by the small curvature radius of the ring wire. Furthermore, going away from the plane of the ring along a line parallell to the wire, we experience a third slow divergence: the field decreases progressively and more and more slowly as we approach the wire. These simple considerations suggest the potential distributions as shown in Fig. 3 (at least in the case of a pure dielectric fluid without free charges). Another advantage of our configuration lies in the fact that the contact area between the metal and the fluid is minimized which can be of prime importance for reducing the induced pollution, particularly in some liquids. In any case, it reasonably allows to use platinum for the opposite electrode the ring or high voltage electrode as we shall often name it in the future - as well as for the heated wire.

3. Mathematical approach The form of the high-voltage electrode invites to deal with the electrostatic problem in a toroidal system of coordinates. We can actually assimilate the wire, which is in fact a very thin cylinder, to the inner part of another torus where the smallest circle of latitude would have the same radius as the real wire and which would belong to the

326

tL W. Vanderschueren et al./Journal of Electrostatics 35 (1995) 323-338

Fig. 4. Relative positions of both toroidal surfaces, r (ring) and w (wire), of the same family.

same family (in this toroidal system) as the external surface of the ring as shown in the Fig. 4. This approximation is of course fully justified in the region of the heart of the system, i.e. the region of the plane of the ring but, wherever we apply the results of the theory, it is worth observing that this approximation is realistic. The increase of the wire diameter, postulated in our treatment, approaches the fact that, in reality, the filament must be fixed to thicker metallic supports. The latter are themselves fixed to the vessel wall which surrounds the whole electrodes set in the same manner, roughly, as the total surface w of Fig. 4. Moreover, given the respective magnitudes of the voltage applied between the ring and the wire, on one hand, and the Joule heating voltage of the wire on the other hand, it is valid for most of the experiments to approximate the ensemble of the wire and the external vessel surface as an equipotential (in practice, the vessel is metallic or it is enclosed in a metallic sheet which protects the cell from parasitic external influences: fields and temperature gradients). Fig. 5 shows both orthogonal circumference patterns in a meridian plane, which are characterized by/~ and r/. By revolution around the x axis, these patterns describe tori (/~) and spheres (q) respectively. Table 1 reproduces definitions and correspondences between toroidal and Cartesian coordinates in a meridian plane. Radii and positions of both circumferences families are also given (cf. for instance, [10, 11] or 1-12]).

H. ~ Vanderschueren et al./Journal o f Electrostatics 35 (1995) 323-338

327

z

I

Fig. 5. Toroidal coordinates disposition and index in a meridian plane (x axis is the symmetry axis). Table 1 Definitions and correspondences between toroidal and Cartesian coordinates ~I =--A'PA

sin X/¢I cosh/~ - cos q sinh # Z/(d cosh/1 -- cos q b/a = coth/~ d/a = cotg t/ b2 _ (,2 ~ a2

~ ln(A'P/AP)

c/a = cosech p e/a = cosec q

All d i s t a n c e s a r e r e f e r r e d t o t h e s a m e l e n g t h a, d i s t a n c e f r o m t h e o r i g i n t o t h e focal p o i n t s A a n d A ' w h e r e all c i r c u m f e r e n c e s o f t h e p a t t e r n r / c o n v e r g e . I n o u r p a r t i c u l a r case, g i v e n t h e t h i n n e s s o f t h e r i n g e l e c t r o d e , t h i s focal d i s t a n c e i d e n t i f i e s itself w i t h t h e r a d i u s o f t h i s e l e c t r o d e .

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328

Basic cell We define here hereafter: ring diameter: ring thickness: wire diameter:

the particular cell we choose as reference for the whole discussion 2br = 30 ram, 2Cr = 1 mm, 2r, = 0.025 mm.

Remark. Among the "classical" curvilinear systems of coordinates, the toroidal one is disadvantaged in the sense that Laplace's equation cannot be solved by the conventional separation of variables. Consequently, equipotential and conjugate surfaces do not merge together with /t and q surfaces, respectively, as one can immediately convince oneself by considering Fig. 6. One can find in [10] general analytical conditions which must be obeyed in order to agree the aforementioned patterns.

4. Potential topography without space charges. Exact solution of the Laplace's equation The general solution of the Laplace's equation AV=0

(1)

in toroidal coordinates is well known (cf. [10] or [12] for instance) and given by

V(p, q, ~0) = x/cosh/~ _ cost/ ~, Vr

~ M(lO.U(q).cb(q9),

(2)

m=O n=O

in which q~ corresponds to the longitude angle and Vr is the value of the applied voltage introduced in order to express the formula in reduced coordinates. Also

M(p) = A,,, P~_ 1/2 (cosh p) + B,,, Q ~_ 1/2 (cosh p),

(3)

N(r/) = C, cos nq + D, sin nq,

(4)

q~(~o) = Em cos m~o + F m sin m~o.

(5)

P,"-1/2 and Qm 1/2 are the half-integer odd Legendre functions. In our particular case, two symmetry conditions lead to an important simplification of the formulae: symmetry of revolution around x axis: ~b(q~) = constant ~ m = 0 - symmetry with respect to the plane x = 0:

N(q) = even function of r / ~ D, = 0.

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329

By including C. and E,. in the coefficients of M(/~), formula (2) reduces to V(p, t/) _ x/cosh # _ cosq ~ [ A , P . _ l / z ( c o s h # ) + B.Q._ 1/:(coshp)]cosnq. (6) Vr n=O The coefficients A, and B. will be determined by the boundary conditions imposed on the applied potential on both extreme tori which delimit the application domain of formula (6). In practice, we work with the grounded wire so that our boundary conditions are ( VIVa)wite = 0,

(71

(V/Vr)ring

(8}

=

1.

The first condition will give the relation between coefficients, and the second will permit their numerical determination. The limiting tori are characterized by their/~ parameter, which can be calculated using Table 1. By applying condition (7) to formula (6), one obtains the next condition which must be verified for all 7: x/cosh/tw -- cost/ ~ [A,P._l/2(cosh#w ) + B.Q,_l/2(coshlaw)]Cosnt/= 0. (9) n=O

This can only be true if all the terms in square brackets, coefficients of non-zero functions of t/, are themselves equal to zero. As a result, one has the relation between A. and B,: Vn ~ A, = - B.

Q.- 1/2 (cosh pw) P , - 1/2 (cosh/~w)"

(10)

By introducing this relation into the potential formula (6), the latter becomes v(~, ~l) = x/cosh ~ - cos t/ ~ B. cos nt/ Vr n=O

X [ Q . - x / 2 ( c o s h # ) - P.-t/2(cosh/~) Q.- x/2(cosh/g,~)] P . - 1/2 (cosh ,Uw)J"

(11)

Applied to the ring potential, in particular, it gives (cosh /A r

-- COS t/)-1/2

=

~ B. cos nt/[Q._ L

1,'2

(cosh/~r)

n=O

-- P n - 1/2 ( c o s h

which must be verified for all r/.

(cosh u t", _,/2 (cos~---h/tw~-)3

~/r)Qn - l / 2

(12)

IL W. Vanderschuerenet al./Journal of Electrostatics 35 (1995) 323-338

330

The second member of this relation is a series of cosnr/. In order to obtain the different B, values, all we need is to expand the first member in a Fourier series and then to identify the coefficients of cos nq. The Fourier series is simply written as (cosh,//r-COS~]) - 1 / 2 =

~

b.cosnq

(13)

n=O

with

bo=~ o(cosh Pr b. = ~

-- COSr/)- i/2 dq,

cos m/(cosh #r - cos t/)-

(141)

1/2 dq.

(142)

The integrals in the second members of (14) are, with the only exception of a factor 2, the generating formulae of the second kind Legendre functions Q._ 1/2. We combine expressions (14) into one formula by introducing the Kronecker symbol (6i~ = 1 for i = j and 0 for i # j): b.-

2 X/~Q, 1 -~o. ~

-1/2(c°shflr)"

(15)

From (12), (13) and (15), one finally obtains for B. the expression

2 x/2[1 B. = 1 + ~o. x

Qn-1/2(coshpw)P.-1/2(coshl~,)] -1. Pn-x/z(COshp.)Qn-1/2(coshpr)J

(16)

By introducing this value in (12), we obtain the general expression (17) of the potential in the space limited by two tori of the family characterized by Pr and Pw in a revolution system and symmetric with respect to r / = 7t or r / = 0: v _ 2 Vr rc

4cosh

- cos

.~

cos nq go.)

1 Qn-1/2(c°sh#w) Pn-1/z(c°sh#)~ ×~--1 -Qn-~l/2(c°sh"~w--) Pn-1/2(c°sh#r)~ Qn-1/2(cosh~).

(17)

5. Important simplification of the potential expression Formula (17) suggests by itself the method of approximation. As we shall prove afterwards, it is not worth the trouble to consider terms of higher order than the first one, provided that a sufficiently thin wire is used to make the ring.

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331

i. 75~ 1.2

."

0.5

""

0.2

o12 o i , 0 i G o i o

""1

i

Fig. 6, Equipotential curves, without free charges (Laplace equation) in an axial plane of the "'basiccell": ring diameter, 2b, = 30 mm; ring thickness, 2cr = 1 mm; wire diameter, 2rw = 0.025 mm

As can be seen in Figs. 10 a n d 11, the second term does n o t exceed 1.6% of the first one for wire a n d ring t h i n n e r t h a n 1 mm. W h e n the ring becomes thicker, the ratio becomes more i m p o r t a n t b u t only far from the wire ( f o r / t > 2). According to this consideration, f o r m u l a (17) simply becomes

V ~ _ x / 2 x/cosh/~ - cost/Ro(/~,Pw,/~r) Vr 7~

(18)

by defining the new derived function

Ro(P, p,~, p~) -

Q - 1/2 (cosh #) - D _ 1/2(# w)P - 1/2 (cosh #) 1

--

D _ 1/2(llw)/D- l/2(/~r)

(19)

with the new p a r a m e t e r s D

1/2 (Pw) ~ Q - u 2 ( c o s h p w ) P - 1/2(cOsh ~w)'

D_ 1/2(]2r) ~---Q - 1/2(cosh jar) P

1/2(cosh/~,) "

(201) (202)

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H.W. Vanderschueren et al./Journal of Electrostatics 35 (1995) 323-338

V/Vrl 0.6 0.4 0.;

Fig. 7. Three-dimensionalplot of the potential in a quadrant of an axial plane of the "basic cell". The main feature of this simplification consists of the fact that (18) can be solved explicitly in cos~ for a fixed ratio v = l//Vr: cos~--cosh~-

~ vRo~(l~,l~w,l~)

.

(21)

Our main purpose is now to "plot" equipotential curves, corresponding to some specific values ofv = V/V~ in an axial plane (i.e. containing the wire axis). Such a work [9] was presented by using traditional tables and integral components of half-integer Legendre functions [11-13], but with a software such as "Mathematica" [14], in which these special functions are implemented, the method becomes simple and really powerful. One can immediately proceed as follows: - from (21), one obtains (for the value v chosen) the limits of g (they correspond of course to the values - 1 and + 1 for cos t/), - for a pair/~, ~ obeying (21), the formulae of Table 1 give x / a and z/a defining a point of the curve to be plotted. Fig. 6 shows the equipotential curves, corresponding to v = 0.3 to 0.9 by steps of 0.1, computed from Eq. (18) in the case of the "basic cell". At a glance, it confirms of course the discrepancy between curves v and curves characterized by a value of/~, as a priori expected in the toroidal system. Fig. 7 shows the potential distribution in the same axial plane.

6. Influence of the modification of the cell parameters It is a characteristic advantage of the analytical solution of a problem to enable to study in depth the implications of parameters modification. It is the purpose of this

H. VK Vanderschueren et al./Journal of Electrostatics 35 (1995) 323-338

333

section (and of the next one) to point out the main characteristic parameter of the cell and to show its effect on the potential distribution. The cell is completely defined by 3 parameters: 2 for the ring (diameter 2br and thickness 2cr) and only one for the wire (diameter 2rw), because the substituted torus belongs to the same family curve as the ring in our toroidal system as we defined it. As formulae (17) and (18) clearly show, the cell parameters interact through D_ 1/2 (Pr) and D _ 1/2 ( # w ), introduced and defined in (20), and thus only through cosh/ar and cosh Pw- The latter are themselves obtainable using Table 1 (with the fact that rw = z when p = P w and r / = 0): cosh

~.L r =

br --,

(221 )

¢r b2

cosh ~w -

2

2 - - Cr +

2

2 rw

2

(222)

b r - - C r -~- F w .

Remark. It is to be noted that here we express cosh #w in terms of br and c~ because they are our practical parameters, but they obey the identity b2 _

c2 ~

a2

so that, the ring being chosen, the poles A and A' of the mathematical characterization (Fig. 5) are dimensionally fixed. This implies that all the graphs we present are expressed in reduced coordinates with reference to "a", the distance between the symmetry axis (axis of the real wire) and the poles and, by way of consequence, that the index w used in (212) is fully justified despite the presence of br and cr in the same formula. Results

1. The homogeneity of formulae (21) shows immediately that there are in fact only two free parameters: if we change all the characteristics in the same proportion, the potential distribution remains isomorphous to itself. Consequently, it is interesting to consider successively the effect of changing the relative thickness of the ring and the wire with regard to the diameter of the ring chosen as reference. Keeping in mind that the whole study is supposed to consider the best compromise (given the standards commonly encountered in every field we are concerned with), a choice of a 30 mm ring is a good one. In that case, we can use a reasonable quantity of liquid in the cell with a sufficient visual field for Schlieren optics observation for instance. 2. In Fig. 8, we report the result of the modification of the wire diameter. It can be seen that this effect is very tenuous: the relative displacement of the equipotential curves corresponds here to a thickness change from 0.050 mm (short dashed lines) to 0.0125 mm (long dashed lines). The latter appears to be a practical limit (such a thin wire is easily broken). 3. In Fig. 9, we report the result of the modification of the ring thickness. In this case, in contrast to the previous one, the relative effect is very important: the

H.W. Vanderschueren et al./Journal of Electrostatics 35 (1995) 323-338

334

1.7'.

1..'

1.,

1.25

1

0.75

0,5

0.

0.

0.3 '

0.2

0.4

0.6

0.8

1

Fig. 8. Influence of a modification of the wire thickness on the position of the equipotential curves in an axial plane of the cell:wire thickness = 25 ~m (basic cell) or 12.5 lam (long dashed curves) or 50 lam (short dashed curves).

Fig. 9. Influence of a modification of the ring thickness on the position of the equipotential curves in an axial plane of the cell:ring thickness = 1 m m (basic cell) or 0.5 m m (long dashed curves) or 2 m m (short dashed curves).

long-dashed lines correspond to a thinner ring (0.5 m m thick) and the short dashed line to a thicker one (2 m m thick) compared to the "basic" one (1 m m thick). The relative thickness of the ring thus appears to be the more important parameter to choose in order to adapt the potential distribution.

7. Validity of the first-order approximation of formula (17) The results of the two previous sections postulate the validity of the limitation of our fundamental formula (17) to its first term. In order to confirm and quantify this approximation, we calculate the order of magnitude of the ratio of the first two terms. This method is powerful in our case where the effect of the parameter q can be disjoint in order to obtain a single general formula. The first term (n = 0) is immediately given by formulae (18)-(20) and the second one (n = 1) can be written in a similar manner (and appearing as a deviation from the exact value): Lz /5 A ' - - -~ Y " x/cosh/~ - cos r/2 cosq R, (p, Pw, P,) Vr--

rc

(18')

H. IV. Vanderschueren et al./Journal o f Electrostatics 35 (1995) 323-338

335 I I

0.015

0.02

iI

0.0125

/s I

0.015

0.01 0.0075

sS SS

0.01

0.005 1. m m

0.005

0.0025

~ 1

2

3

Fig. 10. Influence of a modified wire thickness on the ratio between the second term of the expansion

(n = 1) and the first one (n = 0).

1

4

2

"" " 0 ° 5

'~

i

Fig. 11. Influence of a modified ring thickness on the ratio between the second term of the expansion (n = 1) and the first one (n = 0).

by defining, as before, a new function R t ( # , # w , ~ ) -= Q+ 1/2 ( c ° s h //) - D+ l / 2 ( / . / w ) P + 1/2 ( c o s h /-/) 1 -- D+ 1/2(/,Jw)/D+1/2(/~r)

(19')

with corresponding parameters D+ U2(law) =- Q+ l/2(cosh//w) P+ u2(cosh//w)'

(2o~)

D+l/2(llr ) ~ Q+I/2(cOSh//r) P+ 1 / 2 ( c o s h / / r ) '

(20~)

The ratio mentioned above reduces itself to

A(V/Vr) - : (V/Vr)o

2 cos r/R1 (#, Vw, #~) Ro (/.t, /~/w,/./r)"

(23)

This ratio, always smaller than 2R 1(#, #w, p r)/Ro(#, #w, # r), is characteristic of the cell through pw and #r but depends on the single position parameter #. Figs. 10 and 11 report the results of this majorant calculation in the domain defined by p ~< 4 (i.e. the whole domain for the basic cell in which coshpr = 30) and with the effect of a modification of the wire thickness and the ring thickness, respectively. As a conclusion, limiting the expansion to the first term is adequate provided one does not choose too thick a ring. Just as in the direct study of the influence of a modification of the cell parameters (Figs. 8 and 9), it appears that the main influence concerns the modification of the ring thickness: here on the ratio between the two first terms of the expansion and there on the potential topography.

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H.W. Vanderschueren et aL/Journal of Electrostatics 35 (1995) 323-338

8. Some peculiar remarks about the potential distribution along a ring radius The most important region of the cell is the central area, near the ring plane. It is thus very important to explicate the potential distribution along a ring radius simply obtained by letting ~/= n in formula (18). Fig. 12 shows this distribution in the case of our basic cell. It can be noticed that in both extremes of that domain, the distribution can be easily linearized by an adequate choice of the parameters. Near the wire, the well-known properties of the Legendre functions, corresponding to small values of/~, suggest a logarithmic behaviour which is shown in Fig. 13.

V/Vr1

0.8

0.6

0.4

0.2

-

0 2

,

,

,

•

0.4

,

,

,

|

0.6

,

0~8

z/a

Fig. 12. Potential distribution along a radius of the ring (between wire and ring, where r/= n).

0.4 0.3 0.2 0.1

....

'LS

;2

-:Ls

io.j,.

Fig. 13. Potentialdistribution alonga radius of the ring(between wire and ring, where r/= re)in the vicinity of the wire.

H.W. Vanderschueren et al./Journal of Electrostatics 35 (1995) 323-338

337

V/v r

ii 0.8 0.6 0.4 0.2 12.

fill

1

2

3

4u

Fig. 14. Potential distribution along a radius of the ring (between wire and ring, where r/= ,-t)in the vicinity of the ring.

More peculiar is the behaviour in the vicinity of the ring: one can observe (Fig. 14) that the graph of the potential is reasonably linear when it is expressed as a function of the parameter/~ itself (this remark is general for any value of r/).

Acknowledgements This work has been initiated by R. Coelho when one of the authors (H.W.V.) was preparing his thesis in his laboratory and we are grateful to him for his critical discussion of this manuscript.

References [1] H.Z. Senftleben, Die Einwirkung elektrischer und magnetischer Felder auf das W~irmeleitverm6gen von Gasen, Phys. Zeit., 32 (1931) p. 550. [2] H. Pohl, Dielectrophoresis, Cambridge Univ. Press, Cambridge, 1978. [3] M.F. Haque, E.D. Mshelia and S, Arajs, Effect of electric fields on heat transfer in liquids, J. Phys. D, Appl. Phys., 25 (1992) 740. I-4] R. Coelho and H.W. Vanderschueren, Influence de la purete d'un liquide isolant sur la convection thermique en pr6sence d'un champ ~lectrique, C.R. Acad. Sc. Paris, 274 (1972) 1259. I-5] R. Coelho and H.W. Vanderschueren, Application de l'~lectroconvection et de la strioscopie fi la d~termination rapide de la puret6 des liquides di61ectriques, 6th Imeko World Congress on Measurement and Instrumentation, Dresden, May 1973; Acta Imeko, B424 (1973) 247. 1-6] H.W. Vanderschueren and R. Coelho, On the role of ions and dipoles on electroconvective processes in insulating liquids, 5th Int. Conf. on Conduction and Breakdown in Liquid Dielectrics, Delft, July 1975, Delft Univ. Press, 1975, p. 139. 1-7] R. Coelho and H.W. Vanderschueren, Anomalous electroconvection in weakly conducting liquids, J. Appl. Phys., 48(11) (1977) 4700. [8] H.W. Vanderschueren, Ch. Laurent and J.P. Modolo, Evaluation des additifs antistatiques dans les liquides par thermo61ectroconvection, Rev. G+n. Electr., 7(8) (1985) 585. [9] H.W. Vanderschueren and M.O. Louppe, Topograhie du potentiel et du champ dans une configuration torique, Bull. Sci. A.I.M., (2) (1974) p. 113.

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[-10] E. Durand, Electrostatique, Masson, Paris, 1964. [11] P. Morse and H. Feshbacb, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. [12] L. Robin, Fonctions sph6riques de Legendre et Fonctions sph6roTdales, Coll. C.N.E.T., GauthierVillars, Paris, 1959. [13] P. Byrd and F. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, Berlin, 1971. [14] S. Wolfram, Mathematica, Addison-Wesley, Reading, MA, 1991.